Shape variable de®nition with C0 1 and 2 continuity functions...Shape variable de®nition with C0,...

Shape variable de®nition with C 0, C 1 and C 2 continuity

functions

G. Chiandussi a, G. Bugeda b,*, E. O~nate b

a Department of Mechanical Engineering, Technical University of Turin, Corso Duca degli Abruzzi 24, 10129, Torino, Italyb CIMNE, International Center for Numerical Methods in Engineering, Edi®cio C1, Campus Norte UPC, Gran Capit�an s/n, 08034,

Barcelona, Spain

Received 30 March 1999

Abstract

The present paper proposes a new technique for the de®nition of the shape design variables in 2D and 3D optimisation problems. It

can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the

analytical expression of the CAD de®ning surfaces. The proposed technique allows the surface continuity to be preserved during the

geometry modi®cation process to be de®ned a priori. This capability allows for the de®nition of shape variables suitable for every kind

of discipline involved in the optimisation process (structural analysis, ¯uid-dynamic analysis, crash analysis, aerodynamic analysis,

etc.). Ó 2000 Elsevier Science S.A. All rights reserved.

Keywords: Shape variable de®nition; CAD geometry; Multidisciplinary optimisation; Surface continuity preservation; Finite element

method

1. Introduction

The correct de®nition of shape variables is of the utmost importance in the layout of an optimisation

problem. The identi®cation of the most useful geometry changes is crucial if a satisfactory solution is re-

quired. The shape variable de®nition task has to overcome di�erent di�culties like the selection of the

admissible geometrical changes and the preservation of several geometrical properties as well as geometrical

continuity between ®xed areas and areas subject to deformation.

One of the ®rst researchers to point out the complexity and the di�culties in the shape variable de®nition

was Imam. In his work [1] he identi®es four main techniques for variable de®nition: (1) the independent

node movement technique, (2) the design element technique, (3) the supercurve technique and (4) the shape

superposition technique.

The independent node movement technique is very simple; it uses as shape variables the co-ordinates of

the nodes of the discrete model of the structure.

The design element technique has been extensively used. The design element is a macro ®nite element

consisting of one or more ®nite elements. The isoparametric shape representation, as used for individual

elements, is used here to determine the isoparametric co-ordinates of any point on the surface. This

technique also allows for the determination of the co-ordinates of the node points inside the design element

for mesh generation, which is required every time the shape is changed during optimisation.

www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 188 (2000) 727±742

*Corresponding author.

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 3 5 8 - 8

The third technique mentioned by Imam is the supercurve one. In structured meshes the boundary

surface is fully determined given a few curves laying on it. Only the points on these curves are required to

construct the ®nite element model. In such cases, the parametric representation of curves with a polynomial

expression may be used; the coe�cients of the polynomial expression serve in this case as the shape

variables.

Finally, the shape superposition technique concerns the possibility to superimpose two or more shapes

speci®ed in terms of model location of points on a curve or surface in varying proportions. The linear

combination of these prede®ned shapes allows the generation of a variety of shapes, the coe�cients of the

combination being the shape variables [5]. A similar technique has been used also by Vanderplaats [2] for

airfoil optimisation.

In 1984, Braibant and Fleury [3] proposed a new approach for two-dimensional shape variable de®ni-

tion. The shape description is quite similar to the one based on the design element introduced by Imam. The

region of the structure to be modi®ed during the optimisation process is also de®ned by one or more design

elements that still contain a part of the mesh. Instead of using the shape functions of a two-dimensional

®nite element, blending functions typical of Bezier or B-spline curves are used to determine the co-ordinates

of any point inside the design element or on its boundaries. Therefore the shape variables are no longer the

position of the nodes of an isoparametric two-dimensional element, but the points that control two families

of curves whose cartesian product de®nes the design element. Slope discontinuities at the design element

interfaces can be easily managed. Each boundary of the structure is an automatic piecing of spline of degree

k. The value of k can be selected by the designer and this formulation guarantees the continuity of the

piecing up to the order k ÿ 1 [4].

The ®ve techniques mentioned above have several limitations. Let us consider ®rst the node movement

and the supercurve techniques. The former is practically unusable due to two severe drawbacks: it results in

too many design variables and discontinuity of slope in the shape at the element interfaces takes place in the

®rst few iterations of the process. The latter requires the identi®cation of the parametric equations of

several curves. Several practical applications can be found in two-dimensional problems concerning, for

example, thickness distribution over a shell structure. Moreover, in three-dimensional problems, this

technique requires a non-trivial work that is not compensated by the higher order surfaces that can be

obtained.

These two techniques are not so common and are usually substituted by the curve superposition and the

design element ones. In the former, di�culties arise in the selection and evaluation of the `master' curves or

surfaces whose linear combination will allow identifying the optimal solution. These are usually taken as

the ®rst modal shapes of the structure or the shapes resulting by the application of user-de®ned distributed

loads. Moreover, in this case it is di�cult to control the surface modi®cation of the structure especially if

only local variations are to be considered. The design domain technique is one of the most useful and

practical techniques indeed. It avoids the presence of discontinuities at the element interfaces and allows for

the reduction of the number of the design variables. The limitations of this technique come from the in-

terpolation functions used to extrapolate the design element nodal movements to all nodes present inside

the design element itself. Isoparametric shape functions are used, and because of their origin, they can

guarantee only zero continuity between adjacent elements. So, there is not any possibility to choose the

continuity degree between two adjacent design elements. Moreover, only the nodes inside the design ele-

ment are held in consideration for nodal movement extrapolation. If the design element dimensions are not

large enough, a discontinuity in the movement distribution could be introduced leading to element dis-

tortion and to inaccurate results.

The technique proposed by Braibant and Fleury solves the lack of continuity control at the interface of

two adjacent domains by introducing Bezier or B-spline interpolation functions. This approach can be

easily adopted for two-dimensional problems, but di�culties arise in its extrapolation to three-dimensional

ones. In any case the limitation linked to the movement distribution discontinuity due to the design element

concept as introduced by Imam keep on being present.

All the mentioned techniques are characterised by a common problem: they are based on the discrete

model of the structure analysed.

The present paper proposes a new shape design variable technique. It allows for de®ning the shape

variables directly on the geometry of the analysed structure without any previous knowledge of the ana-

728 G. Chiandussi et al. / Comput. Methods Appl. Mech. Engrg. 188 (2000) 727±742

lytical expression of the CAD surfaces. The design element concept has been introduced and modi®ed

leading to radical simpli®cations. In three-dimensional problems, the domains are de®ned by two-dimen-

sional surfaces whereas, in two-dimensional problems, the domains are de®ned by one-dimensional curves.

Only the `grouping' capability has been preserved. Also the interpolation functions for nodal movement

extrapolation have been modi®ed being de®ned over a lower degree domain. The proposed technique al-

lows for an a priori de®nition of the surface continuity to be preserved during the geometry modi®cation

process. This capability allows for the de®nition of shape variables suitable for every kind of discipline

involved in the optimisation process in structural analysis, ¯uid-dynamic analysis, crash analysis, and

aerodynamic analysis, among others.

2. The method

The proposed method is able to manage two-dimensional as well as three-dimensional shape variable

de®nition problems. The way to operate is exactly the same in the two situations, the only di�erence being

the degree of the geometrical entities. Whereas in a three-dimensional problem the geometry of the analysed

structure is described using two-dimensional surfaces, in two-dimensional problems this is obtained by

using one-dimensional curves.

2.1. Surface modi®cation

Let us consider the three-dimensional geometric model of a structure or a ¯uid domain surrounding an

immersed object. The geometrical model is usually de®ned by several patches linked together. It is possible

to group the patches to de®ne one or more macro-patches. This subdivision allows for the identi®cation of

the portion of the boundary surface of the structure that will be a�ected by shape modi®cations and the one

that will be not. The former can be composed of several macro-patches too. Very often, the mathematical

de®nition of these patches is unknown or it is very complex and outside the scope of the numerical analysis.

In these cases it is impossible to obtain the parametric equation r(u, v) of the macro-patches, and conse-

quently, to evaluate the parametric co-ordinates describing the location of the points belonging to them.

In order to obtain, at least, the approximate parametric co-ordinates of a point belonging to a macro-

patch, a new patch approximating the real one is built by using the information on the co-ordinates of the

vertices of the macro-patch. This de®nes a plane passing through three points if the macro-patch is a tri-

angle, or a curved surface (Coon patch [7,11]) if the macro-patch is a quadrilateral surface. The projection

of every point belonging to the macro-patch over the new one along its normal can now be evaluated. The

projection location over the approximating patch as well as the approximating patch equation can be

transformed from the (x, y, z) reference system to a (u, v) parametric one.

In the proposed approach, the shape variables are de®ned by using vectors applied to the vertices of the

macro-patches previously de®ned. The co-ordinates �x0; y0; z0� of each point belonging to every macro-patch

after shape modi®cation are obtained through the application of the following equation:

�x0; y 0; z0� � �x; y; z� �X

nd

i�1

di �X

nv

j�1

vij �X

np

k�1

rjk�x; y; z�; �1�

where nd is the total number of design variables, nv the total number of vertices of the macro-patch, np the

total number of macro-patches one of whose the vertices is vij, di the value of the ith design variable, vij a

vector related to the design variable i and applied at the vertex j (vector vij is de®ned in direction and

magnitude) and rjk�x; y; z� a function de®ned over the patch approximating the macro-patch k to interpolate

the movement of the vertex vij.

A set of standard interpolation functions with C 0, C 1 and C 2 continuity have been identi®ed. Every

interpolation function is described by using a parametric equation as will be seen in Section 2.2. The

knowledge of the parametric equation of the approximating patch allows to relate it to the standard in-

terpolation function. Consequently, it is possible to identify for every point projection on the approxi-

mating patch a point on the interpolating function describing the effect of the application of a vertex

G. Chiandussi et al. / Comput. Methods Appl. Mech. Engrg. 188 (2000) 727±742 729

movement. The standard interpolation functions are de®ned considering a unit vertex movement. Conse-

quently, the total effect of a non unitary vertex movement can be obtained by a simple multiplication.

Fig. 1 describes this process by using a two-dimensional example for the sake of simplicity. In this case

the macro-patch is a curve and the approximating patch is a straight line connecting the two vertices. The

propagation of the vertex movement on the one-dimensional macro-patch can be described by using four

pictures: (1) the initial situation (a patch with a vector describing the module and direction of the vertex

movement), (2) the projection of the macro-patch points on the approximating patch, (3) the extrapolation

of the vertex movement to all points with reference to a C 0 continuity interpolation function, and (4)

application of the point movement to the original points by using expression (1).

With this approach the new position of each point of a macro-patch is de®ned in terms of the original

geometry plus the superposition of some curved patches. The original geometrical de®nition is taken as a

starting point in the shape modi®cation process and is not needed any more. The shape modi®cations are

controlled in direction and magnitude by the design variables via the vectors applied at the vertices of the

macro-patches.

The design element concept introduced by Imam [1] has been recovered here and used in a slightly

di�erent way. The design element of Imam identi®es a portion of the structural space subject to shape

variation, whereas in the new concept it identi®es only a portion of the boundary surface of the structure

whose shape is modi®ed by the application of a shape variable.

The information introduced with the de®nition of the shape variables needs to be transferred from the

geometrical model to the ®nite element model. The surface modi®cations induced by the design variables

are transformed in a displacement ®eld of the nodes of the discrete model belonging to the macro-patches

(here the term displacement refers to the movement of a node during the shape modi®cation process). To

avoid strong distortion e�ects, it is necessary to propagate the displacement ®eld de®ned at the boundary

surfaces to all internal nodes of the discrete model. Also, in this situation it is necessary to use an inter-

polation function. The absence of speci®c requirements allows for using directly the linear shape functions

typical of the ®nite elements. The movement of the mesh points can now be obtained by computing the

displacements of a ®ctitious linear elastic structure formed by the mesh elements under prescribed boundary

conditions [13].

Fig. 1. The four steps of the propagation of the vertex movement on a one-dimensional macro-patch.


2.2. Boundary movement propagation

The identi®cation of the deformed shape of a macro-patch due to the application of a movement vector

to one of its vertices and the transfer of the information so obtained into a displacement ®eld of the nodes

of the discrete model belonging to it are performed together.

The co-ordinates (x, y, z) of the nodes of the discrete model lying on the surface of the structure are

approximated with the parametric co-ordinates (u, v) of their projection over the approximating patch.

These parametric co-ordinates are used to identify the effect of the application of an imposed displacement

to any vertex of the macro-patch by keeping as a reference a parametric standard surface with parametric

co-ordinates:

06 u; v6 1: �2�

Every reference surface or, more simply, interpolation function r, is characterised by a unit value at one

of the vertices and a null value at all the other ones. Then, for every macro-patch it is possible to identify as

many interpolation functions r as the number of the vertices. The sum of the interpolation functions is

constant and equal to one. This allows us to de®ne the shape variables leading to the displacement of a

portion of the boundary surface parallel to itself.

The interpolation functions r are de®ned to keep certain continuity properties of the boundary surface of

the analysed domain. The C 0 continuity interpolation function is characterised by a null value in corre-

spondence to the sides opposite to the unit value vertex, the C 1 continuity interpolation function has a null

®rst derivative and the C 2 continuity interpolation function has a null second derivative. The smoothness of

the boundary surface of the structure requires continuity of the desired degree to be preserved at the

boundaries between different macro-patches. This implies 0, 1 and 2 cross-boundary continuity degree

respectively for the C 0, C 1 and C 2 continuity interpolation functions. The requirement of unit sum and the

consequent requirement of symmetry lead to ask for cross-boundary derivatives continuous and equal to

zero.

2.2.1. One-dimensional interpolation functions

The identi®cation of the analytical expressions of the interpolation functions in parametric co-ordinates

is quite straightforward in the one-dimensional case. The C 0 interpolation function is simply a ®rst degree

polynomial:

r�u� � u; �3�

the C 1 interpolation function a third degree polynomial:

r�u� � 3u2 ÿ 2u3 �4�

and the C 2 interpolation function a ®fth degree polynomial in the unique parametric co-ordinate u (Fig. 2):

r�u� � 10u3 ÿ 15u4 � 6u5: �5�

They satisfy the requirement of constant unit sum all over the de®nition domain.

2.2.2. Two-dimensional interpolation functions: quadrilateral surfaces

The identi®cation of the interpolation function in the two-dimensional case is more complex. Let us

consider ®rst a quadrangular patch. A curve connecting each pair of adjacent vertices can be identi®ed. The

resulting network of curves identi®es a rectangular patch, which has as its boundaries two u-curves and two

v-curves (Fig. 3). It is assumed that u and v run from 0 to 1 along the relevant boundaries. Then r(u, v)

represents the interior of the surface patch, while r(u, 0), r(1, v), r(u, 1) and r(0, v) represent the four known

boundary curves. The problem of de®ning a surface patch, then, is that of ®nding a suitably well-behaved

function r(u, v) which reduces to the correct boundary curve when u � 0; u � 1; v � 0 or v � 1. This

surface patch can be obtained as boolean sum of two surfaces interpolating the r(u, 0) and r(u, 1) along the v

direction and the r(v, 0) and r(v, 1) along the u direction (Coon patch). The resulting surface patch is

conveniently expressed in the matrix form by [6,13]:


r�u; v� � 1� ÿ uu� �r�0; v�r�1; v�

� �

� r�u; 0� r�u; 1�� 1ÿ v

v

� �

ÿ 1� ÿ uu� �r�0; 0� r�0; 1�r�1; 0� r�1; 1�

� �

�1ÿ v

v

� �

: �6�

Fig. 2. C 0, C 1 and C 2 interpolation functions for the mono-dimensional domain with corresponding analytical expressions.

Fig. 3. Coon patch de®nition through two families of parametric curves.


The auxiliary functions u, (1ÿ u), v and (1ÿ v) are called blending functions, because their effect is to

blend together four separate boundary curves to give a single well-de®ned surface. The surface so identi®ed

has only positional continuity across patch boundaries. The substitution in (6) of the expression of the

boundary curves:

r�u; 0� � 1ÿ u; �7�

r�u; 1� � 0; �8�

r�0; v� � 1ÿ v; �9�

r�1; v� � 0; �10�

r�0; 0� � 1; �11�

r�1; 0� � r�1; 1� � r�0; 1� � 0 �12�

allows us identifying the C 0 continuity interpolation function for quadrilateral patches (Fig. 4)

r�u; v� � �1ÿ u��1ÿ v�: �13�

Looking for a C 1 continuity interpolation function, gradient continuity needs to be preserved and the

patch needs to be de®ned not only in terms of its boundary curves but also in terms of its cross-boundary

slopes. The equation for the patch can be derived in a similar manner to the one previously analysed except

that now it is necessary to use generalised Hermite interpolation rather than generalised linear interpola-

tion. The resulting equation is:

r�u; v� � u0�u� u1�u� w0�u� w1�u��

r�0; v�

r�1; v�

ru�0; v�

ru�1; v�

2

6

6

6

4

3

7

7

7

5

� r�u; 0� r�u; 1� rv�u; 0� rv�u; 1��

u0�v�

u1�v�

w0�v�

w1�v�

2

6

6

6

4

3

7

7

7

5

ÿ u0�u� u1�u� w0�u� w1�u��

r�0:0� r�0; 1� rv�0; 0� rv�0; 1�

r�1; 0� r�1; 1� rv�1; 0� rv�1; 1�

ru�0; 0� ru�0; 1� ruv�0; 0� ruv�0; 1�

ru�1; 0� ru�1; 1� ruv�1; 0� ruv�1; 1�

2

6

6

6

4

3

7

7

7

5

�

u0�v�

u1�v�

w0�v�

w1�v�

2

6

6

6

4

3

7

7

7

5

:

�14�

The substitution of the following curve expressions in (14):

r�u; 0� � 1ÿ 3u2 � 2u3; �15�

r�u; 1� � 0; �16�

r�0; v� � 1ÿ 3v2 � 2v3; �17�

r�1; v� � 0; �18�

r�0; 0� � 1; �19�

r�1; 0� � r�1; 1� � r�0; 1� � 0 �20�


and the imposition of null ®rst and cross derivatives leads to the C 1 interpolation function expression

(Fig. 4):

r�u; v� � �1ÿ 3u2 � 2u3��1ÿ 3v2 � 2v3�: �21�

An identical process is followed for the identi®cation of the C 2 continuity interpolation function. The

equation of the patch with prescribed cross-boundary second derivatives can be derived in a similar way.

Two more blending functions are needed and the square matrix involved will be of order 6� 6. The

function is shown in Fig. 4 and is given by the equation

r�u; v� � �1ÿ 10u3 � 15u4 ÿ 6u5��1ÿ 10v3 � 15v4 ÿ 6v5�: �22�

The requirement of constant unit sum over the domain is satis®ed in the three cases. The interpolation

function de®nition requires verifying the correct sequence of the vertices, independently if clockwise or

anticlockwise. This is to avoid the creation of curves connecting opposite vertices, and then, the creation of

an interpolation function without meaning.

2.2.3. Two-dimensional interpolation functions: triangular surfaces

The de®nition of the interpolation functions for a triangular domain is more complex than the one

previously explained. Let consider the standard triangle with vertices V1(1,0), V2(0,1) and V3(0,0) as in

Fig. 5. Let us de®ne as E1 the side of the triangle opposite to the vertex V1, E2 the side opposite to the vertex

Fig. 4.C 0, C 1 andC 2 interpolation functions for the two dimensional quadrangular domain with corresponding analytical expressions.


V2 and E3 the side opposite to the vertex V3. Moreover, let de®ne P1, P2 and P3 operators that interpolate

the curve representative of each side of the triangle in a direction parallel to the E1, E2 and E3 sides, re-

spectively (Fig. 6).

In the linear case the P1 interpolant can be de®ned as [10]

P1 r u; v� �� ur 1� ÿ v; v� � vr u; 1� ÿ u� �23�

and the P2 interpolant as

P2 r u; v� �� r 0; v� � � r u; 0� � ÿ r 0; 0� �: �24�

The boolean sum of these two interpolants:

P1 � P2 � P1 � P2 ÿ P1P2 �25�

gives the expression of a linear patch over the triangle. With the substitution of the expressions:

r u; 0� � � 1ÿ u; �26�

r 0; v� � � 1ÿ v; �27�

r 1; 0� � � r 0; 1� � � 0; �28�

r 0; 0� � � 1; �29�

describing the side curves and the actual value of the surface at the vertices, it is possible to obtain the C 0

continuity interpolant for the triangle as (Fig. 7)

Fig. 6. Interpolant de®nition for the triangular reference surface.

Fig. 5. Vertex and side de®nition for the triangular reference surface.


r u; v� � � 1ÿ uÿ v: �30�

The approach followed by Gregory [9,12] has been used to identify the interpolation function for the C 1

case. The three interpolants are here de®ned by

P1 r u; v� �� X

1

i�0

ui

v

1ÿ u

� �

� 1� ÿ u�i� r0;i u; 0� � �

X

1

i�0

wi

v

1ÿ u

� �

� 1� ÿ u�i� r0;i u; 1� ÿ u�; �31�

P2 r�u; v�� X

1

i�0

ui

u

1ÿ v

� �

� 1� ÿ v�i� ri;0�0; v� �

X

1

i�0

wi

u

1ÿ v

� �

� �1ÿ v�i� ri;0 1� ÿ v; v�; �32�

P3 r u; v� �� X

1

i�0

ui

u

u� v

� �

� �u� v�i�

o

ou

�

ÿo

ov

�i

r�0; u� v�

�X

1

i�0

wi

u

u� v

� �

� �u� v�i�

o

ou

�

ÿo

ov

�i

r�u� v; 0�; �33�

where the ui and the wi are the Hermite interpolation functions. The expression of the interpolating surface

is given by

r�u; v� � a u; v� �P1 r u; v� �� b u; v� �P2 r u; v� �� c u; v� �P3 r u; v� �� ; �34�

where

a u; v� � � u2 3� ÿ 2u� 6v 1� ÿ uÿ v��; �35�

b u; v� � � v2 3� ÿ 2v� 6u 1� ÿ uÿ v��; �36�

Fig. 7. C 0 and C 1 interpolation functions for the two dimensional triangular domain and corresponding analytical expressions.


a u; v� � � b u; v� � � c u; v� � � 1: �37�

The substitution of (15)±(19) into (34) and the imposition of zero ®rst and cross derivatives, lead to the

expression of the C 1 continuity interpolation function over a triangle (Fig. 7)

r u; v� � �a u; v� �r u; 0� � u� vÿ 1� �

22vÿ u� 1� �

1ÿ u� �3�b u; v� �r 0; v� � u� vÿ 1� �

22uÿ v� 1� �

1ÿ v� �3

� c u; v� �v2 3u� v� �r 0; v� �

u� v� �3

"

�u2 3v� u� �r u; 0� �

u� v� �3

ÿu2v2 ÿ 6u� 6u2 ÿ 6v� 6v2� �

u� v� �4

#

: �38�

The sum of the C 0 and C 1 continuity interpolation functions evaluated at each of the three vertices of the

triangle is constant and equal to one. Due to the symmetry of the equations the sequence of the vertices has

no relevance on the numerical results.

The C 2 continuity interpolation function is still missing and this will be the object of future research

work.

2.3. Displacement propagation inside the discrete model

The interpolation functions previously de®ned allow for the propagation of the displacements de®ned by

the vectors over the macro-patches. The process takes into account only the boundaries of the discrete

model of the analysed domain. To reduce the distortion e�ects that would take place in the areas near the

boundaries of the model during the updating process operated by the optimisation code, it is necessary to

propagate these boundary modi®cations to the internal nodes of the discrete model. These nodes will

belong to the real structure if a structural problem is considered, or to the ¯uid domain if a ¯uid-dynamic

problem is analysed.

The propagation of the boundary displacements can be accomplished by considering the mesh as a

®ctitious linear elastic structure. By solving a linear structural problem with the displacements of the

moving surfaces as imposed displacements, it is possible to obtain the displacement of the nodes of the

entire mesh. By using this procedure, the displacements of the boundaries of the structure are extrapolated

to the entire mesh using the isoparametric shape functions of the ®nite elements. Unfortunately, the so-

lution of the structural problem by considering an isotropic and homogeneous linear material does not

provide a uniform displacement distribution as should be required to minimise the element distortion

during the mesh updating process. In fact, the elements near the moving surfaces are constrained to modify

their shape much more than the ones far from these surfaces.

Finite elements are used in this process just as an interpolation method and, consequently, there is not any

interest about the stress distribution obtained depending on the speci®c material used for the analysis.

Di�erent mechanical properties can therefore be selected and assigned to each mesh element. Assigning

materials with higher mechanical characteristics to the elements near the moving surfaces and materials with

lower mechanical characteristics to the elements far from these surfaces, it is possible to distribute the mesh

deformation more uniformly. Di�erent laws of variation of the mechanical characteristics can be adopted

following geometrical or more physical criteria. For example, following a pure geometrical criterion, the

Young modulus of the mesh elements can be selected depending on the minimum distance of the element

itself from the nearest moving surface. Di�erently, using a more physical approach, the Young modulus can

be selected depending on data concerning the element strain vector or the element strain energy [8].

The best results have been obtained using a distribution law of the material depending on the strain

energy [13]. A preliminary linear analysis with an isotropic and homogeneous material allows for the

evaluation of the strain ®eld for the entire discrete model. Once the strain components e1, e2 and e3 along the

principal strain reference system have been evaluated, it is possible to evaluate the strain energy of each

element as

Ed � �E e21ÿ�

� e22 � e23�

ÿ 2m e1e2� � e2e3 � e3e1��

�39�


and to compare it with the strain energy evaluated considering a constant strain ®eld �e:

Ed � 3�e2E 1� ÿ 2m�: �40�

The comparison leads to the following expression for the new Young modulus to be assigned to each

element for the ®nal analysis:

E ��E

�e2e21 � e22 � e23ÿ �

ÿ 2m e1e2 � e2e3 � e3e1� �

3 1ÿ 2m� �; �41�

where m can be arbitrarily selected and �e has been chosen to be equal to 1.0E)06.

If a ¯uid-dynamic problem is analysed it is necessary to be aware that the surfaces that limit the ¯uid

domain remain ®xed in the solution of the ®ctitious structural problem.

3. Implementation of the method

The proposed method operates by the attribution of well-de®ned properties to several geometrical en-

tities like points, lines or surfaces. So, the geometrical de®nition of the structure to be analysed is needed

only. There is no problem indeed to use the same method for displacement propagation starting from a

discrete model. In this case several facilities as the transfer of the geometrical properties to the discrete

model are lost preserving the main characteristics of the method.

The proposed method operates in two steps. The ®rst step is completely controlled by the user. He has to

identify the most correct and useful design variables to be de®ned taking into account the speci®c kind of

problem to be analysed. The variable identi®cation is performed by the de®nition of several geometrical

properties directly linked to the geometrical entities (points, lines, surfaces) of the geometrical model. It is

necessary to identify: the surfaces in¯uenced by every shape variables (curves in two-dimensional problems)

gathered together in several macro-patches, the vertices of each macro-patch, the displacement vectors

representative of the shape variable applied to the above mentioned vertices and the continuity degree to be

preserved in the propagation of the vertices displacement.

In the second step, all the geometrical information previously de®ned are transferred to the discrete

model and elaborated. A set of instructions is generated by linking every shape variable to the displacement

of the nodes of the discrete model. This information is used by the optimisation code to update the discrete

model at every iteration depending on the values of the design variables. This second phase is completely

independent and does not require any intervention from the user.

The proposed method has been implemented in a speci®c computer program that takes advantage of

the capabilities o�ered by the pre-post-processor system GiD developed at the International Centre of

Numerical Methods in Engineering (CIMNE, Barcelona) [14]. The result of the application of the

program is the creation of two ®les for post processing purposes and one ®le in standard NASTRAN

format containing the relationships between the design variables and the nodal displacements of the

discrete model.

4. Examples

Two examples will be shown next. The ®rst example concerns the shape variable de®nition in an airfoil

¯uid-dynamic optimisation problem. The airfoil is de®ned by several points connected by spline curves. The

¯uid domain has been de®ned as a rectangular `box' around the airfoil itself.

There exist di�erent possibilities for the de®nition of the shape variables: for example, local shape

variables changing only a portion of the geometry of the pro®le can be used or global ones taking in


consideration all the pro®le can be preferred. Three shape variables have been de®ned using the two

di�erent approaches. The local design variables obtained by the application of the three available in-

terpolation functions are shown in Fig. 8. The global design variables obtained by the application of

the three available interpolation functions are shown in Fig. 9; di�erent macro-patches have been used

here to de®ne the three design variables. In the two ®gures, the triangles de®ne the limits of the macro-

patches and the vectors the design variable. The three interpolation facilities have been used to allow

for a comparison between the di�erent results that can be obtained. Mesh modi®cation is quite uniform

and elements near the moving surfaces maintain quite the same dimension and shape as the original

ones. This procedure allows for reducing element distortion e�ect due to surface modi®cation during

the optimisation process.

A three dimensional example concerning the bulb of a ship hull is presented too. Symmetric shape

modi®cations due to seven shape variables are shown in Figs. 10 and 11. The geometry of the bulb and

Fig. 8. Local shape variables for the airfoil problem.

Fig. 9. Global shape variables for the airfoil problem.


its partition in macro-patches can be observed on the left of each ®gure together with the vector

representative of the design variable. Tangent continuity is preserved over all the surface of the structure

by means of the C 1 interpolation function used to extrapolate the vertices displacements. The mesh is

not shown being impossible to identify the singular elements and to keep trace of their shape modi®-

cations.

Again, the quality of the modi®ed shape and the corresponding mesh for each design variable are,

practically, as good as the original one.

Fig. 10. Symmetric surface modi®cation due to seven shape variables de®ned over the bulb of a ship hull.


In a complete optimisation process, all the design variables are used all together, and the shape of each

design is de®ned through the superposition of the shape modi®cation corresponding to each design variable

using expression (1).

5. Conclusions

A new method for shape design variable de®nition has been proposed. It overcomes some of the pitfalls

of the methods formerly proposed by several authors. It is able to manage two-dimensional as well as three-

dimensional problems allowing the user to work directly on the geometrical model of the problem and

simplifying the variable de®nition phase of an optimisation problem layout. C 0, C 1 or C 2 continuity

properties of the boundary surface of the analysed structure can be preserved leading to the de®nition of

shape variables suitable for optimisation problems in structural mechanics as well as in ¯uid-dynamic or

aerodynamic analyses. Boundary surface displacements are propagated over the internal nodes of the mesh

by keeping the distortion of the elements to a minimum. This allows to reduce the error introduced during

the optimisation process due to the presence of highly distorted elements.

Acknowledgements

This work has been supported by the EUMarie Curie Grant BRMA-CT97-5761 held by the ®rst author.

The authors want to acknowledge the support of E. N. BAZAN (Madrid) and of the International Centre

of Numerical Methods in Engineering (CIMNE, Barcelona).

Fig. 11. Continuation of Fig. 10.


References

[1] M.H. Imam, Three dimensional shape optimization, Int. J. Numer. Meth. Engrg. 18 (1982) 661±677.

[2] G. Vanderplaats, Approximation concepts for numerical airfoil optimization, NASA Technical Report no. 1370, 1979.

[3] V. Braibant, C. Fleury, Shape optimal design using B-splines, Comput. Meth. Appl. Mech. Engrg. 44 (1984) 247±267.

[4] V. Braibant, C. Fleury, An approximation-concepts approach to shape optimal design, Comput. Meth. Appl. Mech. Engrg. 53

(1985) 119±148.

[5] S. Zhang, A.P. Belegundu, A systematic approach for generating velocity ®elds in shape optimization, Struct. Optim. 5 (1992)

84±94.

[6] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, New York, 1993.

[7] R.E. Barnhill, R.F. Riesenfeld, Computer Aided Geometric Design, Academic Press, New York, 1974.

[8] G. Chiandussi, G. Bugeda, E. O~nate, A simple method for automatic update of ®nite element meshes, CIMNE Publication no.

147, January 1999.

[9] J.A. Gregory, Smooth interpolation without twist constraints, in: R.E. Barnhill, R.F. Riesenfeld (Eds.), Computer Aided

Geometric Design, Academic Press, New York, 1974.

[10] R.E Barnhill, Smooth interpolation over triangles, in: R.E. Barnhill, R.F. Riesenfeld (Eds.), Computer Aided Geometric Design,

Academic Press, New York, 1974.

[11] S.A. Coons, Surface patches and B-spline curves, in: R.E. Barnhill, R.F. Riesenfeld (Eds.), Computer Aided Geometric Design,

Academic Press, New York, 1974.

[12] R.E Barnhill, J.A. Gregory, Polynomial interpolation to boundary data on triangles, Math. Comput. 29 (131) (1975) 726±735.

[13] G. Chiandussi, G. Bugeda, E. O~nate, Shape variable de®nition with C 0, C 1 and C 2 continuity functions, CIMNE Publication no.

134, July 1998.

[14] GID User Manual, CIMNE, Barcelona.


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